Banks in Cyprus will remain closed at least until Thursday and will then be subject to strict controls to prevent a bank run in the wake of the island's €10bn (£8.5bn) bailout.

All but the country's two biggest banks were slated to open on Tuesday, but the central bank now says all lenders will remain closed to ensure the banking system functions "smoothly". Asked whether Cyprus's banks will reopen on Thursday, Cyprus's finance minister Michalis Sarris said: "Yes, I think they will."

Speaking on Radio 4's Today Programme, Sarris said capital controls will be imposed on Cyprus "for several weeks", restricting the flow of money around the system.

The freezing of the Cypriot banking system follows an international rescue deal that involves restructuring the country's two largest lenders, with heavy losses for wealthy savers. President Nicos Anastasiades acknowledged on Monday that the country had come "a breath away from economic collapse" before its last-minute bailout.

This involved an agreement to radically restructure the country's largest lender, the Bank of Cyprus, and shut down its second largest bank, Laiki, in return for a €10bn bailout from the European Union, the European Central Bank and the IMF.

In a radical departure for a eurozone bailout, depositors in Laiki Bank could lose any savings above €100,000. Bigger savers in Bank of Cyprus will also be forced to contribute to the lender's recapitalisation. Sarris suggested on Monday they could face losses of around 40% on their assets.

After an initial rally on relief that Cyprus had secured a deal, markets took fright when the head of the group of eurozone finance ministers indicated that the rescue could be a template for similar situations.

The euro dropped 1% against the dollar on Monday and remained below $1.29 in early trade on Tuesday. Shares across the eurozone regained some of their losses. The FTSE 100 was up 0.1% and the German DAX was 0.3% higher, but Spain's IBEX was 0.4% lower in early trade.

Jeroen Dijsselbloem, the Dutch finance minister who heads up the euro group, said the relative market calm in recent months, coupled with the lack of market panic following the decision to force private investors and depositors to pay for the bailout of two large Cypriot banks, allowed the eurozone to go after private money more aggressively when banks failed. He later attempted to water down these remarks, indicating that Cyprus was a unique case.

The deal is designed to shrink the Cypriot banking sector, which had grown to eight times the country's €17bn economy. Finance minister Sarris said to Bloomberg on Monday: "We acknowledge that we allowed things in Cyprus to get out of hand."

But he said Cyprus was disappointed by the reaction of some of its eurozone partners, when faced with the potential meltdown of its economy. "This is the European Union. It is supposed to be a community, where you get support from your partners. We got some from some partners; there was a definite hardline from others."

The atmosphere in the negotiating room on Sunday night was, he said, "very, very tense". He added: "We felt under tremendous pressure. When you know that the alternative to reaching a solution would be that your economy is destroyed, that it turns into ruins, you clearly are at a definite disadvantage."

Cypriots had reacted to the agreement with relief as it appeared that at least deposits below €100,000 had been spared the levy. In the streets and cafes of Nicosia, and on TV chat shows, the feeling was that the country had been saved but at a high price. On Tuesday, schoolchildren spilled onto the streets to protest against the bailout deal.

There were warnings the impact could reach beyond Cyprus, particularly with repercussions from Russia, where the prime minister, Dmitry Medvedev, speaking on Monday, said: "They are continuing to steal what has already been stolen." This was a phrase Lenin used to answer the allegation that the Bolsheviks were thieves. Russian officials have repeatedly compared the Cypriot bank levy to Soviet-era expropriation.

"For sure there is anger," said Marios Cosma, head of K Treppides, a firm that serves international clients, mainly rich Russians. "For the first time in Europe, you have a situation where depositors are being called to 'bail in'."

While the Cyprus banking sector has exploded, other countries have even larger banking sectors relative to GDP. Malta's banking sector is relatively larger, and Luxembourg's is more than 20 times its GDP. Malta's finance minister wrote an article in the Malta Times expressing concern about what would happen if it encounters similar problems in the eurozone.

In Cyprus there were calls for a referendum on the bailout package. "It is illegal and undemocratic," said Christos Tombazos, general secretary of the Pancyprian Federation of Labour. "We're talking about massive changes to the banking system. It should go to referendum for the Cypriot people to decide."

Russian officials tried to assuage fears over its nationals' investments in Cyprus. Russian Commercial Bank, a wholly owned subsidiary of state-owned bank VTB, would "not be affected or its losses will be insignificant", said Igor Shuvalov, a deputy prime minister.

The Bank of Cyprus is 9.7% owned by Dmitry Rybolovlev, a Russian based in Monaco whose wealth is estimated at $9.1bn, according to Forbes.

One Russian oligarch, Alexander Lebedev, played down the amount he stood to lose in Cyprus as no more than $10,000. "It's not worth talking about," he said. "Cyprus was always a transit jurisdiction – money would pass through and then go to Lithuania, Latvia, Belize, Switzerland, everywhere. There are plenty of ways [to avoid capital controls], they can split accounts."

The multimillionaire owner of the Evening Standard and Independent expressed doubts that capital controls to be imposed by the Cypriot government in order to stem a bank run would work. "Certain schemes can be put into place," Lebedev said by telephone from Moscow. "This is how Cyprus was making money."